Fully Supersymmetric Hierarchies From A Energy Dependent Super Hill Operator.pdf

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Fully Supersymmetric Hierarchies From A Energy Dependent Super Hill Operator.pdf

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Fully Supersymmetric Hierarchies From A Energy Dependent Super Hill Operator

a r X i v : s o l v - i n t / 9 7 0 2 0 0 2 v 1 1 3 F e b 1 9 9 7 Fully Supersymmetric Hierarchies From A Energy Dependent Super Hill Operator Q. P. Liu?? Departamento de F??sica Teo?rica, Universidad Complutense, E28040-Madrid, Spain. Abstract A super Hill operator with energy dependent potentials is proposed and the associated integrable hierarchy is constructed explicitly. It is shown that in the general case, the resulted hierarchy is multi- Hamiltonian system. The Miura type transformations and modified hierarchies are also presented. ?On leave of absence from Beijing Graduate School, CUMT, Beijing 100083, China. ?Supported by Beca para estancias temporales de doctores y tecno?logos extranjeros en Espan?a: SB95- 1 1 Introduction Schro?dinger equation with energy dependent potential is first studied by Jaulent and Miodek[7] and there in the simplest case, the associated nonlin- ear evolution equations are solved by means of Inverse Scattering Transfor- mation. The problem has been generalized to more general case in[18] and is further shown that the resulted flows are Bi-Hamiltonian system. The remarkable multi-Hamiltonian structures behind have been explored and Miura type maps are obtained in a series papers of Antonowicz and Fordy[1] - [4]. The Lie algebraic reason for constructing Miura map is pro- vided by Marshall [16][17] and this subsequently leads to some new results for the Ito’s system[13]. The most recent result for these hierarchies is their relationship with the zero sets of the tau function of the KdV hierarchy[15]. The generalizations of linear problems with energy potentials are interest- ing and begins with the third order operator or Lax operator for Boussinesq equation[5]. Unlike the Schro?dinger case, one does not have arbitrary poly- nomial dependent expansions here and to have interesting results, one only obtains four cases(see [5] for details). Similarly, Toda system is generalised this way[10]. We notice that integrable s